-1
Pi=TpT 3o= .f
(5-14)
xi and yi were found to be 0 as expected for the aligned position of the fruit. Also as expected, zi
was found to be the negative value of the focal length of the camera and lens (zi = -f).
Assuming that the manipulator position was changed to some position (e1 + dO1, 82 + d%2, d3) by
rotating joints 1 and 2 by some very small increments de1 and d82, the position of the fruit in the
image array was offset from the center by incremental amounts dxi and dyi. Because of the
increments were very small, the second order terms of d81 and d@2 were assumed to be 0. Also,
for small dEi, sin(di) dOi and cos(d)i) = 1. Solving equation 5-11 with the new T3-1 (equation
5-12) for pi yielded
d f de2(d3+z)
z and (5-15)
f dOe S2(d3+z)
dyi=
z (5-16)
Solving for the vision gains resulted in
vxf + 1
K=f z ( and (5-17)
K Y=f S2 d3+1
f +1). (5-18)
The vision gains were approximately proportional to d3 and inversely proportional to z.
Therefore, during a pick cycle, as the picking mechanism was extended toward the targeted
fruit, smaller motions of the two revolute joints were required to compensate for misalignment of
the robot with the targeted fruit. Additionally, Kvy was proportional to the sine of 02. This
term indicated that smaller adjustments in e) were required to compensate for the misalignment
of the robot and the fruit as the position of 02 moved to greater distances from 90.
Another important indication from this kinematic analysis of the relationships among
the position of the joints of the robot, the position of the camera frame, and the position of the